Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-29T12:39:07.358Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuous stochastic games

Published online by Cambridge University Press:  14 July 2016

Matthew J. Sobel
Matthew J. Sobel*
Affiliation:
Yale University
Get access Rights & Permissions [Opens in a new window]

Abstract

Nonzero-sum N-person stochastic games are a generalization of Shapley's two-person zero-sum terminating stochastic game. Rogers and Sobel showed that an equilibrium point exists when the sets of states, actions, and players are finite. The present paper treats discounted stochastic games when the sets of states and actions are given by metric spaces and the set of players is arbitrary. The existence of an equilibrium point is proven under assumptions of continuity and compactness.

Keywords

NONCOOPERATIVE STOCHASTIC GAMEDISCOUNTED MARKOVIAN DECISION PROCESSEQUILIBRIUM POINTDYNAMIC PROGRAMMING
Type
Research Papers
Information
Journal of Applied Probability , Volume 10 , Issue 3 , September 1973 , pp. 597 - 604
Copyright
Copyright © Applied Probability Trust 1973 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research partially supported by N.S.F. Grant GK-13757.

References

[1] Blackwell, D. (1962) Discrete dynamic programming. Ann. Math. Statist. 33, 719–726.10.1214/aoms/1177704593Google Scholar
[2] Blackwell, D. (1965) Discounted dynamic programming. Ann. Math. Statist. 36, 226–235.Google Scholar
[3] Debreu, G. (1952) A social equilibrium existence theorem. Proc. Nat. Acad. Sci. USA 38, 886–893.10.1073/pnas.38.10.886Google Scholar
[4] Denardo, E. V. (1967) Contraction mapping in the theory underlying dynamic programming. SIAM Rev. 9, 165–177.10.1137/1009030Google Scholar
[5] Denardo, E. V. (1971) Markov renewal programs with small interest rates. Ann. Math. Statist. 42, 477–496.Google Scholar
[6] Derman, C. (1965) Markovian sequential control processes-denumerable state space. J. Math. Anal. Appl. 10, 295–302.10.1016/0022-247X(65)90124-1Google Scholar
[7] Dunford, N. and Schwartz, J. T. (1967) Linear Operations, Part I: General Theory. Interscience Publishers Inc., New York.Google Scholar
[8] Harsanyi, J. C. (1967, 1968) Games with incomplete information played by “Bayesian” Players, I–III. Management Sci. 14, 159–182, 302–334, 486–502.Google Scholar
[9] Howard, R. (1960) Dynamic Programming and Markov Processes. Technology Press and Wiley, New York.Google Scholar
[10] Kirman, A. C. and Sobel, M. J. (1971) Dynamic oligopoly with inventories. Unpublished manuscript.Google Scholar
[11] Maitra, A. and Parthasarathy, T. (1970) On stochastic games. J. Opt. Theory and Appl. 5, 289–300.10.1007/BF00927915Google Scholar
[12] Maitra, A. and Parthasarathy, T. (1970) On stochastic games, II. J. Opt. Theory and Appl. 8, 154–160.10.1007/BF00928474Google Scholar
[13] Parthasarathy, K. R. (1967) Probability Measures on Metric Spaces. Academic Press, New York and London.10.1016/B978-1-4832-0022-4.50006-5Google Scholar
[14] Parthasarathy, T. (1972) Discounted, positive, and noncooperative stochastic games. Unpublished manuscript.Google Scholar
[15] Nash, J. (1951) Non-cooperative games. Ann. of Math. 54, 286–295.10.2307/1969529Google Scholar
[16] Radner, R. (1970) Problems in the theory of markets under uncertainty. Technical. Report. No. 3, Collaborative Research on Economic Systems and Organisations, Center for Research in Management Science, University of California, Berkeley.Google Scholar
[17] Rogers, P. D. (1969) Nonzero-sum stochastic games. Report ORC 69–8, Operations Research Center, University of California, Berkeley.Google Scholar
[18] Shapley, L. (1953) Stochastic games. Proc. Nat. Acad. Sci. USA. 39, 1095–1100.10.1073/pnas.39.10.1095Google Scholar
[19] Sobel, M. J. (1971) Noncooperative stochastic games. Ann. Math. Statist. 42, 1930–1935.10.1214/aoms/1177693059Google Scholar